Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds

We study the Smith forms of matrices form f ( C g ) where t , ∈ R [ ] is an elementary divisor domain and companion matrix (monic) polynomial . Prominent examples such are circulant matrices, skew-circulant triangular Toeplitz matrices. In particular, we reduce calculation to that F G quotients by some common divisor. This allows us express last non-zero determinantal as a resultant. A key tool observation ring generated – isomorphic Q = / < > relate several features properties equivalence classes As application let be Alexander torus knot n − 1 calculate By appealing results concerning cyclic branched covers knots cyclically presented groups, this provides homology all Brieskorn manifolds M r s coprime.